Проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту
The problem of constructing an optimal securities portfolio under uncertainty is considered along with the direct and dual problems of fuzzy portfolio optimization. The modified fuzzy portfolio optimization problem is also suggested under a constraint on portfolio volatility. In the dual problem, th...
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System research and information technologies| _version_ | 1867334407490109440 |
|---|---|
| author | Zaychenko, Helen Zaychenko, Yuriy |
| author_facet | Zaychenko, Helen Zaychenko, Yuriy |
| author_institution_txt_mv | [
{
"author": "Helen Zaychenko",
"institution": "Educational and Scientific Complex “Institute for Applied System Analysis” of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv"
},
{
"author": "Yuriy Zaychenko",
"institution": "Educational and Scientific Complex “Institute for Applied System Analysis” of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv"
}
] |
| author_sort | Zaychenko, Helen |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2021-01-19T13:44:38Z |
| description | The problem of constructing an optimal securities portfolio under uncertainty is considered along with the direct and dual problems of fuzzy portfolio optimization. The modified fuzzy portfolio optimization problem is also suggested under a constraint on portfolio volatility. In the dual problem, the portfolio structure is determined, which provides the minimum risk level at the specified profitability level. The use of forecasting share prices for the portfolio model was suggested to support the validity of decisions on the portfolio structure and to reduce the risks. The share price data for the portfolio optimization system are forecasted using Fuzzy Group Method of Data Handling (FGMDH). The experimental studies of the suggested fuzzy models were carried out, and a comparison with the Markowitz model was performed on a stock market. As the result of this work, the foundations of the theory of fuzzy portfolio optimization are built on the basis of the theory of fuzzy sets and an effective forecasting method. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2020.2.07 |
| first_indexed | 2025-07-17T10:26:53Z |
| format | Article |
| fulltext |
H. Zaychenko, Yu. Zaychenko, 2020
88 ISSN 1681–6048 System Research & Information Technologies, 2020, № 2
TIДC
ТЕОРЕТИЧНІ ТА ПРИКЛАДНІ ПРОБЛЕМИ
ІНТЕЛЕКТУАЛЬНИХ СИСТЕМ ПІДТРИМАННЯ
ПРИЙНЯТТЯ РІШЕНЬ
UDC 519.925
DOI: 10.20535/SRIT.2308-8893.2020.2.07
FUZZY PORTFOLIO OPTIMIZATION PROBLEM UNDER
UNCERTAINTY СONDITIONS WITH APPLICATION
OF COMPUTATIONAL INTELLIGENCE METHODS
H. ZAYCHENKO, YU. ZAYCHENKO
Abstract. The problem of constructing an optimal securities portfolio under uncer-
tainty is considered along with the direct and dual problems of fuzzy portfolio opti-
mization. The modified fuzzy portfolio optimization problem is also suggested un-
der a constraint on portfolio volatility. In the dual problem, the portfolio structure is
determined, which provides the minimum risk level at the specified profitability
level. The use of forecasting share prices for the portfolio model was suggested to
support the validity of decisions on the portfolio structure and to reduce the risks.
The share price data for the portfolio optimization system are forecasted using Fuzzy
Group Method of Data Handling (FGMDH). The experimental studies of the sug-
gested fuzzy models were carried out, and a comparison with the Markowitz model
was performed on a stock market. As the result of this work, the foundations of the
theory of fuzzy portfolio optimization are built on the basis of the theory of fuzzy
sets and an effective forecasting method.
Keywords: fuzzy portfolio, modified fuzzy portfolio optimization model, forecast-
ing, share prices, fuzzy GMDH.
INTRODUCTION
The problem of investment in securities had arisen with appearance of the first
stock markets. Careful processing and accounting of investment risks have be-
come an integral and important part of the success of each company. However,
more and more companies have to make decisions under uncertainty, which may
lead to undesirable results. Particularly serious consequences may have the wrong
decisions at long-term investments. Therefore, early detection, adequate and the
most accurate assessment of risk is one of the crucial problems of modern invest-
ment analysis.
The beginning of modern investment theory was put in the article
H. Markowitz, “Portfolio Selection”, which was published in 1952. In this article
mathematical model of optimal portfolio of securities was first proposed. Methods
of constructing such portfolios under certain conditions are based on theoretical
and probabilistic formalization of the concept of profitability and risk. For many
Fuzzy portfolio optimization problem under uncertainty conditions with application
Системні дослідження та інформаційні технології, 2020, № 2 89
years the classical theory of Markowitz was the main theoretical tool for optimal
investment portfolio construction, after which most of the novel theories were
only modifications of the basic theory.
New approach in the problem of investment portfolio construction under un-
certainty is connected with fuzzy sets theory, created about half a century ago in
the fundamental work of Lotfi Zadeh [6]. By using fuzzy numbers in the forecast
share prices a decision-making person was not required to form probability esti-
mates.
The application of fuzzy sets technique enables to create a novel theory of
fuzzy portfolio optimization under uncertainty and risk deprived of drawbacks of
classical portfolio theory.
The main source of uncertainty is changing stock prices of securities at the
stock market as the decision on portfolio is based on current stock prices while
the implementation of portfolio is performed in future and portfolio profitability
depends on future prices which are unknown at the moment of decision making.
Therefore that to raise the reliability of decision concerning portfolio and cut
possible risk it’ s needed to forecast future prices of stocks. For this the applica-
tion of inductive modeling method, so-called Fuzzy Group Method of Data Han-
dling (FGMDH) seems to be very perspective.
The goals of this paper are to present and analyze the results in fuzzy portfo-
lio optimization theory, to consider and analyze so-called classic and new direct
problems of fuzzy portfolio optimization, to estimate the application of FGMDH
for stock prices forecasting and to carry out experimental investigations for esti-
mation of the efficiency of the elaborated theory.
DIRECT PROBLEM OF FUZZY PORTFOLIO OPTIMIZATION
Problem statement
Let us consider a share portfolio of N components and its expected behaviour at
time interval ],0[ T . Each of a portfolio component Ni ,1 at the moment T is
characterized by its financial profitability ir (evaluated at a point T as a relative
increase in the price of the asset for this period) Assume the capital of the inves-
tor be equal 1. The problem of share portfolio optimization consists in a finding of
a vector of share prices distribution in a portfolio }{ ixx Ni ,1 maximizing the
expected income at the set risk level.
In process of practical application of Markowitz model its drawbacks were
detected [2, 3]:
The hypothesis about normality of profitability distributions usually isn’t
proved in practice.
Stationarity of price processes is not always valid in practice.
At last, the risk of stocks is considered as a covariance or its volatility,
therefore the decrease in profitability in relation to the expected value, and prof-
itability increase are estimated in this model absolutely equally, while for the
owner of securities these events are absolutely different.
These weaknesses of Markowitz theory caused necessity of development es-
sentially new approach for definition of an optimum investment portfolio.
H. Zaychenko, Yu. Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2020, № 2 90
Let’s consider the main principles and ideas of a fuzzy method for portfolio
optimization [1, 2, 3, 4].
The risk of a portfolio is not its volatility (covariance), but possibility that
expected profitableness of a portfolio will appear below some pre-established
planned value r*.
Correlation of stock prices in a portfolio is not considered and not taken into
account.
Profitability of each security is not random, but fuzzy number. Similarly, re-
striction on extremely low level of profitability can be both usual scalar and fuzzy
number of any kind. Therefore, optimize a portfolio in such statement may mean,
in that specific case, the requirement to maximize expected profitability of a port-
folio in time moment T at the fixed risk level.
Profitability of a security on termination of ownership term is expected to be
equal r and lies in a certain range.
For i -th security let’s denote:
ir — the expected profitability of the i-th security;
ir1 — the lower border of profitability of the i-th security;
ir2 — the upper border of of the i-th security;
),,( 21 iiii rrrr — profitability of the i-th security is a triangular or Gaus-
sian fuzzy number.
Then profitability of a portfolio is equal:
N
i
ii
N
i
ii
N
i
ii rxrrxrrxrr
1
2max
11
1min ;; , (1)
where ix is the weight of i -th security in a portfolio ( its portion), for which the
following conditions holds
N
i
ix
1
1, 10 ix . (2)
Critical level of profitability of a portfolio at the moment of T given by an
investor may be non-fuzzy or fuzzy number ),,( *
2
**
1
* rrrr as well.
Fuzzy-set portfolio optimization with triangular membership functions
To determine the structure of a portfolio which provides the maximum profitability
at the set risk level, it is required to solve the following problem [2, 3].
Find }{}{ opt XX , for which maxr , under condition const)( X ,
where r is a portfolio profitability, is a desired risk, vector X satisfies (2).
In [2, 3] the following expression for risk β (x) was obtained for critical value r*
N
i
ii
N
i
ii
N
i
iiN
i
ii
N
i
iiN
i
ii
N
i
ii rxrx
rrx
rrxrxr
rxrx
1
1
1
*
1*
11
1
*
1
1
1
2
ln
1
(3)
Fuzzy portfolio optimization problem under uncertainty conditions with application
Системні дослідження та інформаційні технології, 2020, № 2 91
Taking into account also that profitability of a portfolio is equal to:
N
i
ii
N
i
ii
N
i
ii rxrrxrrxrr
1
2max
11
1min ;; ,
where ),,( 21 iii rrr is the profitability of i-th security, we obtain the following op-
timization problem [2, 3]:
;max
1
N
i
iirxr (4)
const ; (5)
Nixx i
N
i
i ,1,0,1
1
. (6)
At a risk level variation 3 cases are possible: 0 , 1 and 10 .
Consider in detail the last case 10 .
This case is possible when
N
i
ii
N
i
ii rxrrx
1
*
1
1 or when
*
1
rrx
N
i
ii
N
i
iirx
1
2 .
Assume
N
i
ii
N
i
ii rxrrx
1
*
1
1 .Then using (3) the problem (4)–(6) is reduced
to the following nonlinear programming problem [2–5]:
max
1
N
i
iirxr . (7)
Under conditions
N
i
ii
N
i
ii
N
i
iiN
i
ii
N
i
iiN
i
ii
N
i
ii rxrx
rrx
rrxrxr
rxrx
1
1
1
*
1*
11
1
*
1
1
1
2
ln
1
; (8)
*
1
1 rrx
N
i
ii
; (9)
*
1
rrx
N
i
ii
; (10)
Nixx i
N
i
i ,1,0,1
1
. (11)
Let
N
i
ii
N
i
ii rxrrx
1
2
*
1
. Then the problem (9)–(11) is reduced to the fol-
lowing nonlinear programming problem [2–5]:
max
1
N
i
iirxr ; (12)
H. Zaychenko, Yu. Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2020, № 2 92
N
i
iirxr
1
1
*
N
i
ii
N
i
ii
N
i
ii
N
i
ii
N
i
iiN
i
ii
rxrxrxrx
rxr
rxr
1
1
1
2
1
1
1
2
1
*
1
* 1
*ln* ; (13)
*
1
2 rrx
N
i
ii
; (14)
*
1
rrx
N
i
ii
; (15)
Nixx i
N
i
i ,1,0,1
1
. (16)
The penalty functions algorithm of minimization is suggested to find the so-
lution of problems (7)–(11) and (12)–(16). Assume both problems (7)–(11) and
(12)–(16) be solvable. Then structure of a required optimumal portfolio will be
such vector NixX i ,1}{ being the solution of problems (7)–(11) or
(12)–(16) for which the criterion value will be greater.
Modified fuzzy portfolio optimization under constraint on volatility
Analysis of fuzzy portfolio optimization model (7)–(10) and previous experimen-
tal results [2–5] has shown that the constraint on risk in fuzzy portfolio model
doesn’t make real restriction of profitability and acts in the same way as criterion:
the less risk the greater is profitability.
It leads to some bias in portfolio content, usually the portion of the most
profitable share takes the greatest value and practically doesn’t change with risk
variation. Therefore it was suggested to modify fuzzy portfolio model by intro-
ducing the constraint on portfolio volatility :giv
;)(
1
giv12
N
i
iii rrx
where
max)( min)(:var 1212givgiv , iiii rrrr .
Using this transformation we obtain two linear programing problems pre-
sented below:
а) for the case ,~
1
1
1
*
i
i
ii
i
i rxrrx
NN
LP-problem takes the following form:
N
i
ii rxr
1
max;~~
Fuzzy portfolio optimization problem under uncertainty conditions with application
Системні дослідження та інформаційні технології, 2020, № 2 93
;)(
1
giv12
N
i
iii rrx
;*
1
1
rrx i
i
i
N
;~ *
1
rrx i
N
i
i
;1 ,1,0,
1
Nii i
N
i
xx
;]max)( min)[( 1212giv rrrr
b) for the case *
2
1 1
,
N N
i i i i
i i
x r r x r
we obtain the following
LP-problem:
;max~~
1
i
N
rxr
i
i
;)(
1
giv12
N
i
iii rrx
*
2
1
;
N
i i
i
x r r
*
1
;
N
i i
i
x r r
,,1,0,
1
1 Nii i
N
i
xx
where
.]max)( min)([ 212giv irrrr
If both these LP-problems are solvable then the structure optimal invest
portfolio Nixx i ,1}{ will be determined for which the portfolio profitability
will be greater.
THE ANALYSIS AND COMPARISON OF THE RESULTS BY MARKOWITZ
AND FUZZY-SETS MODELS
For comparative analysis of investigated methods the following corporations
Apple INC, Cisco, Intel, Microsoft, Visa. were chosen whose shares are presented
in stock- exchange NYSE and used for Dow Jones index calculation. The infor-
mation about share prices of these corporations were taken from site
https://www.nyse.com. The data about share prices were taken in the period from
01.01.2018 to 29.06.2018 year, in a whole the period of 125 days.
H. Zaychenko, Yu. Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2020, № 2 94
The obtained data from NYSE contained the information about opening and
closing share prices of the chosen companies.
For fuzzy portfolio optimization the calculated profitability range (region)
and expected profitability value were previously calculated for each share. In
Markowitz model expected profitability of a share is calculated as a mean
}{rMm and risk of an asset is considered as a dispersion of the profitability
value.
Using this data for all stocks the following indicators were calculated:
1) mean profitability, 2) left bound of profitability minr , 3) right bound maxr
4) profitabilty range ( volatility) Δзад and variance 2 . These data are presented
in the table 1.
T a b l e 1 . Shares Profitability, volatility, and variance
N Company , %r minr maxr Δgiv 2σ
1 APPLE INC 0,0735 -0,9585 0,8700 1,8285 1,6015
2 CISCO SYS INC 0,0932 -0,8681 0,9564 1,8245 1,7895
3 INTEL CORP 0,0643 -1,3545 1,1903 2,5348 2,8340
4 MICROSOFT CORP 0,1183 -0,9063 0,8227 1,7290 1,3972
5 VISA INC 0,1255 -0,8200 0,6976 1,5176 1,1478
Portfolio optimization for Markowitz model
Construct mathematical model of Markowitz using (7)–(11). Number of shares in
a portfolio equals to 5. Choose the initial risk level 5,0 . Then at each step
increasing risk by 0,5 consider and solve models for risk value from 0,5 to 4,5.
For its solution penalty functions method was applied and software kit in program
language C# was developed. The obtained results for risk values )5,45,0(
with step 0,5 are presented in the table 2.
T a b l e 2 . Portfolio constructed by Markowitz model for different risk levels
Shares
N
Risk
β
P
ro
fi
ta
b
il
it
y
A
P
PL
E
IN
C
C
IS
C
O
S
Y
S
IN
C
IN
T
E
L
C
O
R
P
M
IC
R
O
S
O
FT
C
O
R
P
V
IS
A
IN
C
1 0,5 0,0928 0,2491 0,1491 0,0745 0,0745 0,3745
2 1,0 0,0940 0,2529 0,1529 0,0767 0,0764 0,3764
3 1,5 0,0969 0,2581 0,1513 0,0833 0,0830 0,3830
4 2,0 0,0981 0,2643 0,1643 0,0908 0,0908 0,3908
5 2,5 0,0988 0,2635 0,1635 0,0862 0,0862 0,3862
6 3,0 0,1005 0,4296 0,0294 0,1125 0,0875 0,3875
7 3,5 0,1022 0,3852 0,0852 0,1137 0,0862 0,3862
8 4,0 0,1022 0,3852 0,0852 0,1137 0,0862 0,3862
9 4,5 0,1022 0,3852 0,0852 0,1137 0,0862 0,3862
Fuzzy portfolio optimization problem under uncertainty conditions with application
Системні дослідження та інформаційні технології, 2020, № 2 95
The corresponding dependence “profitablility/risk” for Markowitz model
based on this data is presented in the fig. 1.
r
2σ
Fig. 1. Dependence profitability versus risk for optimal portfolio by Markowitz
Fuzzy portfolio optimization model
As it was stated previously for fuzzy portfolio with risk variation there are three
different cases. Consider the case 0 < β < 1 when
i
i
ii
i
i rxrrx
NN ~
1
1
1
*
.
Let solve fuzzy portfolio problem for this case. The corresponding expres-
sion for risk )(X is presented in (8). As critical value for risk take 08,0* r .
The following model is non-linear programing problem. For its solution pen-
alty functions method was applied. This problem was solved using developed
software kit in language C# for risk values )9,01,0(: with step 0,1 and the
results are presented in the table 3.
T a b l e 3 . Optimal portfolios for fuzzy model for different risk values
Assets
N
Risk
β
P
ro
fi
ta
b
il
it
y
A
P
PL
E
I
N
C
C
IS
C
O
S
Y
S
IN
C
IN
T
E
L
C
O
R
P
M
IC
R
O
SO
F
T
C
O
R
P
V
IS
A
I
N
C
1 0,1 0,0983 0,2406 0,1343 0,0750 0,1375 0,3750
2 0,2 0,0973 0,2355 0,1355 0,0722 0,1361 0,3722
3 0,3 0,0959 0,2307 0,1307 0,0696 0,1348 0,3696
4 0,4 0,0945 0,2261 0,1261 0,0672 0,1336 0,3672
5 0,5 0,0932 0,2271 0,1217 0,0648 0,1324 0,3648
6 0,6 0,0904 0,2154 0,1154 0,0577 0,1288 0,3577
7 0,7 0,0893 0,2116 0,1116 0,0558 0,1279 0,3558
8 0,8 0,0882 0,2079 0,1079 0,0539 0,1269 0,3539
9 0,9 0,0872 0,2045 0,1045 0,0522 0,1261 0,3522
H. Zaychenko, Yu. Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2020, № 2 96
The corresponding dependence profitability/risk for fuzzy model is pre-
sented in the fig. 2.
r
Fig. 2. Dependence “optimal profitability versus risk for fuzzy portfolio”
From this figure it follows that with risk value increase the profitability of
optimal fuzzy portfolio drops. Compare these results with Markowitz model (fig. 1).
Dependences of expected profitability on degree of the portfolio risk,
received by the above specified methods, are practically opposite. The reason of
such result is the different understanding of a portfolio risk. In the fuzzy-set
method the risk is understood as a situation when expected profitability of a
portfolio falls below the critical level r*set by an investor, so with decrease of
expected profitability risk of portfolio investments to be less than the critical
value, increases .
In Markowitz model the risk is considered as the degree of expected
profitability variability of a portfolio, in both cases of smaller and greater
profitability that contradicts the common sense. The various understanding of
portfolio risk level is also the reason of difference of a portfolio structure,
received by different methods.
The structures of an optimum portfolio which are obtained by both methods
for the same risk levels are also quite different.
New fuzzy portfolio optimization model with constraint on volatility
In this problem portfolio volatility was calculated as difference between maximal
and minimal profitability for all the shares. In order to implement this approach
the data of all shares prices in portfolio in the period since 01.02.2018 to 02.08. 2018
were taken and for all shares the mean profitability, low and upper bounds and
volatility were calculated. The obtained results are presented in the table 4.
T a b l e 4 . Shares profitability and volatility
N Company , %r minr maxr Δ
1 APPLE INC 0,0373 -1,0950 0,9000 1,9550
2 CISCO SYS INC 0,3982 -1,1214 1,0653 2,1867
3 INTEL CORP 0,1865 -1,5118 1,4362 2,9481
4 MICROSOFT CORP 0,2428 -0,8635 0,7402 1,6037
5 VISA INC 0,1969 -1,0242 0,7914 1,8156
Fuzzy portfolio optimization problem under uncertainty conditions with application
Системні дослідження та інформаційні технології, 2020, № 2 97
From analysis of the results in the table 4 ift follows that the greatest mean
profitability have shares of CISCO SYS INC, and the least volatility is for shares
of MICROSOFT CORP.
After solution of the corresponding LP-problems optimal portfolio profitability
in dependence of volatility was determined which is presented in the table 5.
T a b l e 5 . Dependence of mean portfolio profitability versus volatility
Shares
N giv
P
or
tf
ol
io
vo
la
ti
li
ty
A
P
P
L
E
I
N
C
C
IS
C
O
S
Y
S
IN
C
IN
T
E
L
C
O
R
P
M
IC
R
O
SO
FT
C
O
R
P
V
IS
A
I
N
C
1 1,7 0,269 0,000 0,165 0,000 0,835 0,000
2 1,8 0,295 0,000 0,337 0,000 0,663 0,000
3 1,9 0,322 0,000 0,508 0,000 0,492 0,000
4 2,0 0,349 0,000 0.680 0,000 0.320 0,000
5 2,1 0,376 0,000 0,851 0,000 0,149 0,000
6 2,2 0,3982 0,000 1,000 0,000 0,000 0,000
Using these results the chart of portfolio profitability for optimal portfolio is
shown in the fig. 3.
giv
r
Fig. 3. Dependence of profitability versus volatility for optimal portfolio
From this chart it follows that with increase of volatility the mean profitability
nof fuzzy portfolio also grows. As we see this dependence well matches to
Markowitz portfolio model.
THE DUAL PORTFOLIO OPTIMIZATION PROBLEM
Fuzzy portfolio optimization
Now consider the portfolio optimization problem dual to the problem
(7)–(11) [7].
H. Zaychenko, Yu. Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2020, № 2 98
To minimize )(x ; under conditions
*
1
rrxr
N
i
ii
;
N
i
ix
1
1, 0ix ,
where expression for )(x is given by (3)
Optimality Conditions for Dual Fuzzy Portfolio Problem
As it was proved in [8] function )(x is convex therefore the dual portfolio prob-
lem (14)–(16) is a convex programming problem. Taking into account that con-
straints (16) are linear compose Lagrangian function:
1)()(),,(
11
*
N
i
i
N
i
ii xrxrxxL .
The optimality conditions by Kuhn–Tucker are such:
Nir
x
x
x
L
l
ii
1,0
)(
;
01,0
1
*
1
N
i
i
N
i
li x
L
rrx
L
,
and conditions of complementary slackness
,1,0 Nix
x
L
i
i
0,0,0*
1
i
N
i
ii xrrx
L
,
where 0 and are indefinite Lagrange multipliers.
This problem may be solved by standard methods of convex programming,
for example method of feasible directions or method of penalty functions.
THE APPLICATION OF FGMDH FOR STOCK PRICES FORECASTING IN
PORTFOLIO OPTIMIZATION PROBLEM
Consider the results of experimental investigations of the developed models and
methods of fuzzy portfolio optimization. The profitability of leading companies at
NYSE in the period from 03.09.2013 to 17.01.2014 were used as the input data in
experimental investigations. The companies included: Canon Inc. (CAJ), McDon-
ald's Corporation (MCD), PepsiCo, Inc. (PEP), The Procter & Gamble Company
(PG), SAP AG (SAP).
For forecasting it was suggested to use fuzzy GMDH method with triangular
membership functions, linear partial descriptions, training sample of 70%, fore-
casting for 1 step.
The fuzzy GMDH allows to construct forecasting model using experimental
data automatically without participation of an expert. Besides, it may work un-
der uncertainty conditions with fuzzy input data or data given as intervals [9].
Fuzzy portfolio optimization problem under uncertainty conditions with application
Системні дослідження та інформаційні технології, 2020, № 2 99
The next profitability values real and forecasted by fuzzy GMDH on 17.01.2014
were obtained (table 6):
T a b l e 6 . The profitability of shares on date 17.01.2014 (real and fore-
casted), %
Profitability
Companies Real
value
Low
bound
Forecasted
value
Upper
bound
MSE test
sample
MAPE test
sample
CAJ -1,270 -1,484 -1,246 -1,008 2,2068 0,0295
MCD -0,105 -0,347 -0,118 0,111 2,5943 0,0091
PEP 0,206 0,001 0,242 0,483 3,0179 0,0177
PG 0,162 0,041 0,170 0,299 1,6251 0,0197
SAP 0,843 0,675 0,867 1,059 2,3065 0,0164
Thus, as the result of application of FGMDH the shares profitability values
were forecasted to the end of 20-th week (17.01.2014).
Let the critical profitability level be 0,7%. Varying the risk level we obtain
the following results at the end of 20-th week (17.01.2014) for triangular MF. The
results are presented in the table 7 and the dependence “optimal profitability
versus risk” is shown in the fig. 4.
T a b l e 7 . Parameters of the optimal portfolio with critical level %,*r 70
Low bound Expected profitability Upper bound Risk level
0,55133 0,74591 0,94049 0,2
0,53462 0,72954 0,92446 0,25
0,51544 0,71084 0,90624 0,3
0,51894 0,71431 0,90968 0,35
0,5045 0,70018 0,89587 0,4
0,50877 0,70425 0,89973 0,45
0,522 0,71731 0,91262 0,5
0,50197 0,69752 0,89308 0,55
0,46358 0,66014 0,8567 0,6
Fig. 4. Dependence of the expected portfolio profitability versus risk level for triangular MF
H. Zaychenko, Yu. Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2020, № 2 100
As one can see in the fig. 4, the dependence profitability - risk has descend-
ing type, the greater is the risk the lesser is profitability which is opposite to clas-
sical probabilistic method. When the expected profitability decreases, the risk
grows.
The profitability of the real portfolio is 0,7056%.This value falls in calcu-
lated corridor of profitability for optimal portfolio [0,58944, 0,78264, 0,97584]
built with application of forecasting, indicating the high accuracy of the forecast
(see table 7).
Let’s consider the results obtained by solving the dual problem using trian-
gular MF. In this case, the investor sets the rate of return and the problem is to
minimize the risk.
The optimal portfolio is presented in table 8 and the dependence “critical
level of return- risk” is shown in the fig. 5.
T a b l e 8 . Parameters of the optimal portfolio (dual task)
Low bound
Expected
profitability
Upper bound Risk level
Critical rate
of return
0,58944 0,78264 0,97584 0,00025 0,6
0,59846 0,79141 0,98437 0,01468 0,65
0,61478 0,80735 0,99991 0,04973 0,7
0,6229 0,81531 1,00772 0,13347 0,75
0,63606 0,82822 1,02037 0,26399 0,8
0,64945 0,84181 1,03417 0,49937 0,85
0,63712 0,82933 1,02153 0,72631 0,86
0,63382 0,82612 1,01843 0,8333 0,87
0,62559 0,81805 1,01052 0,91214 0,88
Fig. 5. Dependence of the risk level on a given critical return *r
From these results one can see that the curve “dependence risk – given criti-
cal level of profitability” has the ascending character, because with the growth of
the critical value of profitability r* the probability that the expected profitability
would be lower than a given critical value increases.
Fuzzy portfolio optimization problem under uncertainty conditions with application
Системні дослідження та інформаційні технології, 2020, № 2 101
CONCLUSION
The problem of optimization of the investment portfolio under uncertainty is con-
sidered in this paper. The fuzzy-set approach for solving the direct and dual port-
folio optimization problems was suggested and explored. The modified direct
portfolio problem with constraint on volatility was suggested and investigated
The results of solving the problems were presented. The optimal portfolios for the
five assets at NYSE stock market were constructed and analyzed.
The problem of stock prices forecasting for portfolio optimization was also
investigated. Inductive modelling method fuzzy GMDH was applied for stocks
prices forecasting at NYSE stock market in the problem of fuzzy portfolio
optimization. The application of fuzzy GMDH enables to decrease risk of the
wrong decisions concerning portfolio content.
After analysis of experiments it was detected that the dependence
“profitableness – risk” in the basic fuzzy model has descending type, the greater
risk the lesser is profitableness that is opposite to classical probabilistic
Markowitz model. In the modified fuzzy portfolio problem the dependence
“profitable volatility” has ascending type like Markowitz model.
The dependence “risk versus given critical level of return” has ascending
type, because with the growth of the critical level of profitability the probability
that the expected profitability appears to be lower than a given critical value increases.
As the main result of this research the fundamentals of theory of fuzzy
portfolio optimization under uncertainty have been developed free from
drawbacks of the classic portfolio theory.
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2006.
Received 02.07.2020
H. Zaychenko, Yu. Zaychenko
ISSN 1681–6048 System Research & Information Technologies, 2020, № 2 102
INFORMATION ON THE ARTICLE
Yu. P. Zaychenko, ORCID: 0000-0001-9662-3269, Educational and Scientific Complex
“Institute for Applied System Analysis” of the National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: zaychenkoyuri@ukr.net.
H.Yu. Zaychenko, ORCID: 0000-0002-4546-0428, Educational and Scientific Complex
“Institute for Applied System Analysis” of the National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: syncmaster@bigmir.net.
ПРОБЛЕМА НЕЧІТКОЇ ПОРТФЕЛЬНОЇ ОПТИМІЗАЦІЇ В УМОВАХ НЕВИЗНА-
ЧЕНОСТІ З ВИКОРИСТАННЯМ МЕТОДІВ ОБЧИСЛЮВАЛЬНОГО ІНТЕЛЕКТУ /
О.Ю. Зайченко, Ю.П. Зайченко
Анотація. Розглянуто проблему побудови оптимального портфеля з цінних
паперів в умовах невизначеності, а також пряму та двоїсту задачі портфельної
оптимізації. Запропоновано нову постановку задачі нечіткої оптимізації порт-
феля за обмежень на волатильність. У двоїстій задачі визначається структура
портфеля, яка забезпечує мінімум ризику за обмежень на заданий рівень до-
хідності. Для підвищення обгрунтованості рішень щодо структури портфеля та
зменшення ризику запропоновано використання прогнозування цін акцій
в моделі портфеля. Дані за цінами акцій прогнозуються з використанрням не-
чіткого МГУА. Проведено експериментальні дослідженрня запропонованих
нечітких моделей та порівняння з моделлю Марковітца на ринку цінних папе-
рів. У результаті роботи створено основи теорії нечіткої портфельної оптимі-
зації на базі теорії нечітких множин та методу прогнозування.
Ключові слова: нечіткий портфель, модифікована модель нечіткої портфель-
ної оптимізації, прогнозування цін акцій, нечіткий МГУА.
ПРОБЛЕМА НЕЧЕТКОЙ ПОРТФЕЛЬНОЙ ОПТИМИЗАЦИИ В УСЛОВИЯХ
НЕОПРЕДЕЛЕННОСТИ С ПРИМЕНЕНИЕМ МЕТОДОВ ВЫЧИСЛИТЕЛЬНОГО
ИНТЕЛЛЕКТА / Е.Ю. Зайченко, Ю.П. Зайченко
Аннотация. Рассмотрены проблема построения оптимального портфеля из
ценных бумаг в условиях неопределенности, а также прямая и двойственная
задачи нечеткой портфельной оптимизации. Предложена новая постановка за-
дачи нечеткой оптимизации портфеля с ограничением на волатильность порт-
феля. В двойственной задаче определяется структура портфеля, которая обес-
печивает минимум риска при заданном уровне доходности. Для обеспечения
обоснованности принимаемых решений по структуре портфеля и уменьшения
риска предложено использовать прогнозирование цен акций в модели портфе-
ля. Данные по ценам акций для системы портфельной оптимизации прогнози-
руются с использованием нечеткого МГУА. Проведены экспериментальные
исследования предложенных нечетких моделей и сравнение с моделью Мар-
ковитца на рынке ценных бумаг. В результате работы построены основы тео-
рии нечеткой портфельной оптимизации на основе теории нечетких множеств
и эффективного метода прогнозирования
Ключевые слова: нечеткий портфель, модифицированная модель нечеткой
портфельной оптимизации, прогнозирование цен акций, нечеткий МГУА.
|
| id | journaliasakpiua-article-216232 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:26:53Z |
| publishDate | 2020 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/ad/52201eb7ed0e93e5d238784a7944e7ad.pdf |
| spelling | journaliasakpiua-article-2162322021-01-19T13:44:38Z Fuzzy portfolio optimization problem under uncertainty conditions with application of computational intelligence methods Проблема нечеткой портфельной оптимизации в условиях неопределенности с применением методов вычислительного интеллекта Проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту Zaychenko, Helen Zaychenko, Yuriy fuzzy portfolio modified fuzzy portfolio optimization model forecasting share prices fuzzy GMDH нечіткий портфель модифікована модель нечіткої портфельної оптимізації прогнозування цін акцій нечіткий МГУА нечеткий портфель модифицированная модель нечеткой портфельной оптимизации прогнозирование цен акций нечеткий МГУА The problem of constructing an optimal securities portfolio under uncertainty is considered along with the direct and dual problems of fuzzy portfolio optimization. The modified fuzzy portfolio optimization problem is also suggested under a constraint on portfolio volatility. In the dual problem, the portfolio structure is determined, which provides the minimum risk level at the specified profitability level. The use of forecasting share prices for the portfolio model was suggested to support the validity of decisions on the portfolio structure and to reduce the risks. The share price data for the portfolio optimization system are forecasted using Fuzzy Group Method of Data Handling (FGMDH). The experimental studies of the suggested fuzzy models were carried out, and a comparison with the Markowitz model was performed on a stock market. As the result of this work, the foundations of the theory of fuzzy portfolio optimization are built on the basis of the theory of fuzzy sets and an effective forecasting method. Рассмотрены проблема построения оптимального портфеля из ценных бумаг в условиях неопределенности, а также прямая и двойственная задачи нечеткой портфельной оптимизации. Предложена новая постановка задачи нечеткой оптимизации портфеля с ограничением на волатильность портфеля. В двойственной задаче определяется структура портфеля, которая обеспечивает минимум риска при заданном уровне доходности. Для обеспечения обоснованности принимаемых решений по структуре портфеля и уменьшения риска предложено использовать прогнозирование цен акций в модели портфеля. Данные по ценам акций для системы портфельной оптимизации прогнозируются с использованием нечеткого МГУА. Проведены экспериментальные исследования предложенных нечетких моделей и сравнение с моделью Марковитца на рынке ценных бумаг. В результате работы построены основы теории нечеткой портфельной оптимизации на основе теории нечетких множеств и эффективного метода прогнозирования. Розглянуто проблему побудови оптимального портфеля з цінних паперів в умовах невизначеності, а також пряму та двоїсту задачі портфельної оптимізації. Запропоновано нову постановку задачі нечіткої оптимізації портфеля за обмежень на волатильність. У двоїстій задачі визначається структура портфеля, яка забезпечує мінімум ризику за обмежень на заданий рівень дохідності. Для підвищення обгрунтованості рішень щодо структури портфеля та зменшення ризику запропоновано використання прогнозування цін акцій в моделі портфеля. Дані за цінами акцій прогнозуються з використанрням нечіткого МГУА. Проведено експериментальні дослідженрня запропонованих нечітких моделей та порівняння з моделлю Марковітца на ринку цінних паперів. У результаті роботи створено основи теорії нечіткої портфельної оптимізації на базі теорії нечітких множин та методу прогнозування. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2020-09-25 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/216232 10.20535/SRIT.2308-8893.2020.2.07 System research and information technologies; No. 2 (2020); 88-102 Системные исследования и информационные технологии; № 2 (2020); 88-102 Системні дослідження та інформаційні технології; № 2 (2020); 88-102 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/216232/219291 Copyright (c) 2021 System research and information technologies |
| spellingShingle | нечіткий портфель модифікована модель нечіткої портфельної оптимізації прогнозування цін акцій нечіткий МГУА Zaychenko, Helen Zaychenko, Yuriy Проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту |
| title | Проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту |
| title_alt | Fuzzy portfolio optimization problem under uncertainty conditions with application of computational intelligence methods Проблема нечеткой портфельной оптимизации в условиях неопределенности с применением методов вычислительного интеллекта |
| title_full | Проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту |
| title_fullStr | Проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту |
| title_full_unstemmed | Проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту |
| title_short | Проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту |
| title_sort | проблема нечіткої портфельної оптимізації в умовах невизначеності з використанням методів обчислювального інтелекту |
| topic | нечіткий портфель модифікована модель нечіткої портфельної оптимізації прогнозування цін акцій нечіткий МГУА |
| topic_facet | fuzzy portfolio modified fuzzy portfolio optimization model forecasting share prices fuzzy GMDH нечіткий портфель модифікована модель нечіткої портфельної оптимізації прогнозування цін акцій нечіткий МГУА нечеткий портфель модифицированная модель нечеткой портфельной оптимизации прогнозирование цен акций нечеткий МГУА |
| url | https://journal.iasa.kpi.ua/article/view/216232 |
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